Polynomial Entropy for Interval Maps and Lap Number

Abstract

We prove an upper bound for the polynomial entropy of continuous, piecewise monotone maps of the interval, according to the number of intervals of monotonicity of its iterates. We give examples that show that this inequality is sharp. As a direct consequence of this inequality, the polynomial entropy of monotone, continuous, interval maps is always less than or equal to one. We give examples where we can also obtain lower bounds. We also prove analogous inequality in terms of total variations of the iterates of these interval maps. Also, this inequality is sharp.